Thread: Proof for some exponent rules

1. Proof for some exponent rules

I'm looking for proof for the following exponent rules:

$\displaystyle x^px^q = x^{p+q}$
$\displaystyle x^p/x^q = x^{p-q}$
$\displaystyle x^py^p = (xy)^p$

Without using any logarithmic functions and not through induction.

2. Here's a way to look at it. I hope it's allowed.

$\displaystyle x^{p}x^{q}=\underbrace{x\cdot{x}\cdot{x}\cdot{x}\c dot{x}....\cdot{x}}_{\text{p factors of x}}\underbrace{\cdot{x}\cdot{x}\cdot{x}\cdot{x}... ...\cdot{x}}_{\text{q factors of x}}$

Since the total number of factors of x on the right is p+q, this is equal to

$\displaystyle x^{p+q}$

Will that suffice?.

I'm not quite sure that would qualify as a proof. Wouldn't it count more as induction?

4. Originally Posted by Spec

I'm not quite sure that would qualify as a proof. Wouldn't it count more as induction?
i think its valid. wouldn't really describe it as an induction

5. The proof is okay. You can do it with induction if you really want, but there is no point. It is simple enough itself.

6. hmmmmmmm
Maybe you should throw in some values (positive and negative) as that would certainly classify as proof.

1a) Let x = 1, p = 2, q = 3
1²*1³ = 1, 15 = 1 therefore
Let x = -2, p = 2, q = 3
-2²*-2³ = -32, -25 = -32 therefore

1b) Let x = 1, p = 2, q = 3
1²/1³ = 1, 1-1 = 1 therefore
Let x = -2, p = 2, q = 3
-2²/-2³ = -0.5, -2-1 = -0.5 therefore

1c) Let x = 1, y = 2, p = 3
1³*2³ = 8, (1*2)³ = 8 therefore
Let x = -2, y = 3, p = 3
-2³*3³ = -216, (-2*3)³ = -216 therefore

If you prefer you can use your own values as long as one is negative and one is positive. This would definitely class as proof as far as i can see.
Hope this helps.

7. Originally Posted by Sean12345
hmmmmmmm
Maybe you should throw in some values (positive and negative) as that would certainly classify as proof.

1a) Let x = 1, p = 2, q = 3
1²*1³ = 1, 15 = 1 therefore
Let x = -2, p = 2, q = 3
-2²*-2³ = -32, -25 = -32 therefore

1b) Let x = 1, p = 2, q = 3
1²/1³ = 1, 1-1 = 1 therefore
Let x = -2, p = 2, q = 3
-2²/-2³ = -0.5, -2-1 = -0.5 therefore

1c) Let x = 1, y = 2, p = 3
1³*2³ = 8, (1*2)³ = 8 therefore
Let x = -2, y = 3, p = 3
-2³*3³ = -216, (-2*3)³ = -216 therefore

If you prefer you can use your own values as long as one is negative and one is positive. This would definitely class as proof as far as i can see.
Hope this helps.
i'm afraid citing specific examples does not constitute proof, at least not as far as mathematics is concerned

8. The proof I gave is satisfactory.

9. I always thought that was an induction Galactus.